Abstract
We study the automorphisms \phi of a finitely generated
free group F. Building on the train-track
technology of Bestvina, Feighn and Handel, we provide a topological
representative f:G --> G of a power of \phi that behaves very much
like the realization on the rose of a positive automorphism. This
resemblance is encapsulated in the Beaded Decomposition Theorem
which describes the structure of paths in G obtained by repeatedly
passing to f-images of an edge and taking subpaths.
This decomposition is the key
to adapting our proof of the quadratic isoperimetric inequality for
F \rtimes_\phi Z, with \phi positive, to the general case.
To illustrate the wider utility of our topological normal form, we provide a
short proof that for every w in F, the function n |--> |\phi^n(w)|
grows either polynomially or exponentially.